Question: Determine how many solutions exist for the system of equations. ${5x+y = 10}$ ${-10x-2y = 20}$
Solution: Convert both equations to slope-intercept form: ${5x+y = 10}$ $5x{-5x} + y = 10{-5x}$ $y = 10-5x$ ${y = -5x+10}$ ${-10x-2y = 20}$ $-10x{+10x} - 2y = 20{+10x}$ $-2y = 20+10x$ $y = -10-5x$ ${y = -5x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x+10}$ ${y = -5x-10}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.